We study the boundedness of a mutation class for quivers with real weights. The main result is a characterization of bounded mutation classes for real quivers of rank 3.
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Keywords: quiver mutation, real weights, rank 3 quivers
CC-BY 4.0
Casals, Roger; Ke, Kenton. Boundedness criteria for real quivers of rank 3. Algebraic Combinatorics, Volume 9 (2026) no. 2, pp. 403-425. doi: 10.5802/alco.481
@article{ALCO_2026__9_2_403_0,
author = {Casals, Roger and Ke, Kenton},
title = {Boundedness criteria for real quivers of rank~3},
journal = {Algebraic Combinatorics},
pages = {403--425},
year = {2026},
publisher = {The Combinatorics Consortium},
volume = {9},
number = {2},
doi = {10.5802/alco.481},
language = {en},
url = {https://alco.centre-mersenne.org/articles/10.5802/alco.481/}
}
TY - JOUR AU - Casals, Roger AU - Ke, Kenton TI - Boundedness criteria for real quivers of rank 3 JO - Algebraic Combinatorics PY - 2026 SP - 403 EP - 425 VL - 9 IS - 2 PB - The Combinatorics Consortium UR - https://alco.centre-mersenne.org/articles/10.5802/alco.481/ DO - 10.5802/alco.481 LA - en ID - ALCO_2026__9_2_403_0 ER -
%0 Journal Article %A Casals, Roger %A Ke, Kenton %T Boundedness criteria for real quivers of rank 3 %J Algebraic Combinatorics %D 2026 %P 403-425 %V 9 %N 2 %I The Combinatorics Consortium %U https://alco.centre-mersenne.org/articles/10.5802/alco.481/ %R 10.5802/alco.481 %G en %F ALCO_2026__9_2_403_0
[1] Cluster-cyclic quivers with three vertices and the Markov equation, Algebr. Represent. Theory, Volume 14 (2011) no. 1, pp. 97-112 | DOI | MR | Zbl
[2] A binary invariant of matrix mutation, 2023 (forthcoming, J. Comb. Algebra) | arXiv | DOI | Zbl
[3] Trigonometric Diophantine equations (On vanishing sums of roots of unity), Acta Arith., Volume 30 (1976) no. 3, pp. 229-240 | DOI | MR | Zbl
[4] Mutation Cycles from Reddening Sequences, 2025 | arXiv | Zbl
[5] Geometry of mutation classes of rank 3 quivers, Arnold Math. J., Volume 5 (2019) no. 1, pp. 37-55 | DOI | MR | Zbl
[6] Mutation-finite quivers with real weights, Forum Math. Sigma, Volume 11 (2023), Paper no. e9, 22 pages | DOI | MR | Zbl
[7] Quiver mutations, 2022 https://www.samuelfhopkins.com/... (transparencies for a talk at the Open Problems in Algebraic Combinatorics conference at the University of Minnesota, May 16-20, 2022)
[8] Cyclically ordered quivers, 2024 | arXiv | Zbl
[9] Long mutation cycles, Selecta Math. (N.S.), Volume 31 (2025) no. 5, Paper no. 103, 44 pages | DOI | MR | Zbl
[10] Introduction to cluster algebras. Chapters 1–3, 2016 | arXiv | Zbl
[11] Double Bruhat cells and total positivity, J. Amer. Math. Soc., Volume 12 (1999), pp. 335-380 | DOI | MR | Zbl
[12] Cluster algebras I, J. Amer. Math. Soc., Volume 15 (2002) no. 2, pp. 497-529 | DOI | MR | Zbl
[13] Cluster algebras II, Invent. Math., Volume 154 (2003) no. 1, pp. 63-121 | DOI | MR | Zbl
[14] On the approximate periodicity of sequences attached to non-crystallographic root systems, Exp. Math., Volume 27 (2018) no. 3, pp. 265-271 | DOI | MR | Zbl
[15] Discrete dynamical systems from real valued mutation, Exp. Math., Volume 33 (2024) no. 2, pp. 261-275 | DOI | MR | Zbl
[16] Mutation-acyclic quivers are totally proper, 2024 | arXiv | Zbl
[17] Congruence invariants of matrix mutation, J. Pure Appl. Algebra, Volume 229 (2025) no. 3, p. Paper No. 107920, 14 | DOI | MR | Zbl
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